It is shown that an electron avalanche travels with a velocityv_ that is dependent upon Townsend's ionization coefficient &agr; and the diffusion coefficient for the electronsD_. Specifically,v_=&mgr;_E+&agr;D_, where &mgr;_ is the electron mobility, and the applied fieldEis assumed constant. Numerical machine calculations have been made to study the effect of diffusion and secondary ionization at the cathode in pulsed Townsend discharges. Illustrative examples show the utility of these calculations in measurements of the Townsend ionization coefficients as well as electron and ion mobilities. It is shown that the diffusion‐broadened avalanche for high values of &agr;d(dis the gap distance) requires more than the electron transit timed/v_ to make a gap crossing.